Financial Risk Manager Part 1

Explanation

This question requires application of the omitted variables formula in regression analysis. Given the two single-variable regression models:

• $\hat{Y} = 0.5767 + 0.5633X_1$ (omitting $X_2$)
• $\hat{Y} = 0.6767 - 0.7633X_2$ (omitting $X_1$)

And the covariance/variance information:

• $\text{Cov}(X_1, X_2) = 0.603$
• $\text{Var}(X_1) = 0.874$
• $\text{Var}(X_2) = 0.75$

Step 1: Calculate the omitted variable bias coefficients

The omitted variable bias formula states that when $X_2$ is omitted from the regression:

$\hat{\beta}_1^{simple} = \beta_1 + \beta_2 \cdot \delta_1$

Where $\delta_1 = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_1)}$

Similarly, when $X_1$ is omitted:

$\hat{\beta}_2^{simple} = \beta_2 + \beta_1 \cdot \delta_2$

Where $\delta_2 = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_2)}$

Step 2: Calculate the delta values

$\delta_1 = \frac{0.603}{0.874} = 0.6899$ $\delta_2 = \frac{0.603}{0.75} = 0.804$

Step 3: Set up the system of equations

From the first simple regression: $0.5633 = \beta_1 + \beta_2 \cdot 0.6899$

From the second simple regression: $-0.7633 = \beta_2 + \beta_1 \cdot 0.804$

Step 4: Solve for $\beta_1$ and $\beta_2$

Solving this system:

• $\beta_1 = 2.4476$
• $\beta_2 = -2.7312$

Step 5: Calculate the intercept

The intercept in the multiple regression is: $\hat{\alpha} = \bar{Y} - \beta_1 \bar{X}_1 - \beta_2 \bar{X}_2$

From the regression equations, we can derive that: $\hat{\alpha} = \hat{Y}_i - 2.4476X_{1i} + 2.7312X_{2i}$

This matches option D exactly.

Key Insight: The omitted variable bias formula allows us to recover the true coefficients from simple regressions when we know the covariance structure between the explanatory variables._